Solutions - 5110-05

(1) Define random variables:

Let $X_3\sim \text{Bin}(3,p)$ represent the car with three components.

Let $X_5\sim \text{Bin}(5,p)$ represent the car with five components.

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(2) Find probabilities that both cars work:

For the three-component car, we want $P[X_3\ge 2]$, so

$$P[X_3\ge 2]=\sum _{k=2}^3\left(\genfrac{}{}{0px}{}{3}{k}\right)p^k{(1-p)}^{3-k}\gg \gg 3p^2-2p^3$$

For the five-component car, we want $P[X_5\ge 3]$, so

$$P[X_5\ge 3]=\sum _{k=3}^5\left(\genfrac{}{}{0px}{}{5}{k}\right)p^k{(1-p)}^{5-k}\gg \gg 10p^3-15p^4+6p^5$$

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(3) Find when $P[X_3\ge 2]>P[X_5\ge 3]$:

Solve the inequality for $p$:

$$\begin{array}{cc}3p^2-2p^3& >10p^3-15p^4+6p^5\\ 6p^5-15p^4+12p^3-3p^2& <0\\ 3p^2(2p^3-5p^2+4p-1)& <0\\ 3p^2{(p-1)}^2(2p-1)& <0\gg \gg 0<p<\frac{1}{2}\end{array}$$